Abstract
AbstractWe prove the mixed‐norm ‐boundedness of a general class of singular integral operators having a multi‐parameter singularity and acting on vector‐valued (UMD Banach lattice‐valued) functions. Moreover, families of such operators with uniform assumptions are shown to be not only uniformly bounded but ‐bounded, a genuinely stronger property that is often needed in applications. Previous results of this nature only dealt with convolution‐type or slightly more general paraproduct‐free singular integrals. In contrast, our analysis specifically targets the array of different partial paraproducts that arise in the multi‐parameter setting by interpreting them as paraproduct‐valued one‐parameter operators. This new point‐of‐view provides a conceptual simplification over the existing representation results for multi‐parameter operators, which is a key to the proof of the boundedness of these operators.
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